Non-semibounded sesquilinear forms and left-indefinite Sturm-Liouville problems

Fleige, Andreas

doi:10.1007/bf01203080

Cited by 16 publications

(15 citation statements)

References 14 publications

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“…There are also results on the case where r>0 and s is indefinite, cf. [2,4,7,12,14,15]. Allegretto and Mingarelli [1] give an analogue of (1.4) (with r + s replaced by rs + ) for a special case (q=0, r>0, 1Âr # L ) which is transformed into one to which (1.4) applies directly.…”

Section: Introductionmentioning

confidence: 98%

Existence and Asymptotics of Eigenvalues of Indefinite Systems of Sturm–Liouville and Dirac Type

Binding¹,

Volkmer²

2001

Journal of Differential Equations

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Existence and asymptotic behavior of eigenvalues of indefinite Sturm Liouville and Dirac systems with integrable coefficients are investigated using Pru fer angle analysis. The results generalize those previously obtained by Atkinson, Mingarelli, Gohberg and Krein and are based on a new theorem that determines the asymptotic behavior of the solution of a Riccati-type equation containing a large parameter. Academic Press

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Section: Introductionmentioning

confidence: 98%

Existence and Asymptotics of Eigenvalues of Indefinite Systems of Sturm–Liouville and Dirac Type

Binding¹,

Volkmer²

2001

Journal of Differential Equations

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“…For details see [Fle99,Lemma 6]. Recently, based on [GKMV13], more general operators of the form div(C∇·) with Dirichlet boundary conditions and indefinite Hermitian coefficient matrices C have been investigated [HKKS12].…”

Section: The M C Intosh Conditionmentioning

confidence: 99%

A generalisation of the form method for accretive forms and operators

Elst

Sauter

Vogt

2015

Journal of Functional Analysis

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Abstract. The form method as popularised by Lions and Kato is a successful device to associate m-sectorial operators with suitable elliptic or sectorial forms. M c Intosh generalised the form method to an accretive setting, thereby allowing to associate m-accretive operators with suitable accretive forms. Classically, the form domain is required to be densely embedded into the Hilbert space. Recently, this requirement was relaxed by Arendt and ter Elst in the setting of elliptic and sectorial forms.Here we study the prospects of a generalised form method for accretive forms to generate accretive operators. In particular, we work with the same relaxed condition on the form domain as used by Arendt and ter Elst. We give a multitude of examples for many degenerate phenomena that can occur in the most general setting. We characterise when the associated operator is m-accretive and investigate the class of operators that can be generated. For the case that the associated operator is m-accretive, we study form approximation and Ouhabaz type invariance criteria.

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“…[4,5,6,7] use the name "regular" instead of "hyper-solvable". We use a different terminology because we do not want to confuse hyper-solvable forms with Θ-regular forms (see Remark 3.10).…”

Section: The Second Representation Theoremmentioning

confidence: 99%

“…More precisely, in [4,5,8,15] it is proved that, if D = D(|T | In this paper we adapt Kato's second representation theorem to a solvable sesquilinear form Ω (not necessarily symmetric), represented by an operator T , and with domain D = D(|T | where T = U |T | is the polar decomposition of T . The paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

“…McIntosh [12], Fleige et al [4,5], Grubisić et al [8] and Schmitz [15] adapted the second representation theorem for symmetric sesquilinear forms they considered. More precisely, in [4,5,8,15] it is proved that, if D = D(|T | In this paper we adapt Kato's second representation theorem to a solvable sesquilinear form Ω (not necessarily symmetric), represented by an operator T , and with domain D = D(|T | where T = U |T | is the polar decomposition of T .…”

Section: Introductionmentioning

confidence: 99%

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A Kato's second type representation theorem for solvable sesquilinear forms

Corso

2018

Journal of Mathematical Analysis and Applications

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Abstract. Kato's second representation theorem is generalized to solvable sesquilinear forms. These forms need not be non-negative nor symmetric. The representation considered holds for a subclass of solvable forms (called hyper-solvable), precisely for those whose domain is exactly the domain of the square root of the modulus of the associated operator. This condition always holds for closed semibounded forms, and it is also considered by several authors for symmetric sign-indefinite forms. As a consequence, a one-to-one correspondence between hyper-solvable forms and operators, which generalizes those already known, is established.

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Non-semibounded sesquilinear forms and left-indefinite Sturm-Liouville problems

Cited by 16 publications

References 14 publications

Existence and Asymptotics of Eigenvalues of Indefinite Systems of Sturm–Liouville and Dirac Type

Existence and Asymptotics of Eigenvalues of Indefinite Systems of Sturm–Liouville and Dirac Type

A generalisation of the form method for accretive forms and operators

A Kato's second type representation theorem for solvable sesquilinear forms

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